Problem 33 - Algebra 1

Question

Will choosing the two points that are the farthest apart always give you the closest line? Explain why or why not. If not, sketch a counterexample.

Step-by-Step Solution

Answer

No, a line between two points that are the farthest apart will not always result in the closest or shortest line. This is because the line of shortest distance between any two points is a straight line connecting them directly, and if two points are the farthest from each other, that connecting line will be the longest rather than the shortest. The triangle inequality theorem backs up this idea. A counterexample would be three points, A, B and C, with B and C being the farthest apart but the line from A to B or A to C being shorter than the line from B to C.

Step 1: Conceptual understanding

First, understand the initial proposition: If two points that are the farthest apart in a plane are chosen, will the line connecting them always be the closest, or shortest, line? 'Closest' here would mean the line of shortest distance. In a plane with multiple points, the distance between any two points is the straight line that connects them. If two points are the farthest from each other, the line connecting them would actually be the longest distance between any two points in the plane. Therefore, the initial proposition is not true.

Step 2: Theoretical basis: Triangle inequality theorem

The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. This provides a theoretical basis for our argument that a line connecting two points that are not the farthest apart might be shorter than the line connecting the points that are the farthest apart.

Step 3: Counterexample

To provide a counterexample, imagine three points in a plane, A, B and C. Assume that B and C are the farthest apart, and A is somewhere between them. In this case, the line between B and C is longer than the line between A and B, or A and C. This shows that choosing the two points that are the farthest apart does not always yield the 'closest' or shortest line, contradicting the initial proposition.

Key Concepts

Distance Between Points

Understanding the distance between two points is crucial in geometry. To find the distance between any two points in a plane, you use the distance formula. This formula is particularly helpful when you have the coordinates of the points. The distance formula is derived from the Pythagorean theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(d\) represents the distance between the points.

By applying this formula, you can determine the straight-line distance between two points on a plane. The straight line is indeed the shortest path between two points. Hence, if any point is positioned such that the direct line forms the sides of a triangle, the sum of the other sides will always be longer (by the Triangle Inequality Theorem), further explaining the problem's solution.

Counterexample

A counterexample is a specific case for which a general statement is proved false. In our exercise, to counter the proposition that connecting the two farthest points always results in the shortest line, we use a geometric configuration. Imagine three points A, B, and C on a plane. Suppose points B and C have the greatest distance between them. Intuitively, one might believe that a line segment connecting B and C is the shortest possible line, but this intuition can be misleading. If we introduce a third point, A, located somewhere between points B and C but not directly on the line connecting them, we can find a shorter path:

• The direct line from A to B or A to C might be shorter than B to C.
• This effectively disproves the initial claim by illustrating that B to C, being the farthest apart, does not yield the shortest path compared to exploring other connections.

This simple yet powerful example clarifies how assumptions can sometimes be countered with logical, contrary illustrations in geometry.

Plane Geometry

Plane geometry is the study of figures on a flat, two-dimensional surface. It's the foundational realm where various shapes and distances are calculated and analyzed. In plane geometry, we deal with basic shapes like points, lines, and circles, and their characteristics such as angles and distances. The concepts of plane geometry are crucial because they lay the groundwork for understanding more advanced geometrical theorems and properties.

• Shapes and their properties: Plane geometry introduces the fundamental building blocks of shapes - triangles, rectangles, circles, and more.
• Interrelationships: It explores how these shapes overlap, intersect, and relate, providing a basis for proofs and logical reasoning in the field.

By mastering the basics of plane geometry, students gain insight into complex geometrical principles, including the Triangle Inequality Theorem used in the exercise to analyze points' distances effectively.